Biomedical Engineering Reference
In-Depth Information
A propagation model of the beams in the interfaces was developed to also account for the
effect of birefringence. Approximate values are available for the following quantities:
thickness of the microscope slide (
t
s
) and the measured values of
p
pm
and
p
sm
. From the
latter values and from
Eq. (17.16)
, it is possible to compute
sm
. The optical paths
of the beams from the prism to the emergence in the saline solution can hence be
determined. At this point, one must add the Maxwell
φ
pm
and
φ
Neumann equations that relate
stresses to the observed retardations:
n
o
2
n
g
5
A
σ
1
B
σ
1
2
(17.17)
n
e
2
n
g
5
B
σ
1
1
A
σ
2
The above equations also contain the constants
A
and
B
and the residual compressive
stresses
2
present in the prism. Approximate values for A and B are also known.
The utilization of the above equations to formulate an optimization problem leads to the
best values that define the posed problem. The set of values listed in
Table 17.1
were
reported in Refs.
[13,17]
. The optimization process was started from three different initial
points in order to converge to the global optimum: (i) lower bounds of design variables
(Run A); (ii) mean values of design variables (Run B); (iii) upper bounds of design
variables (Run C).
σ
1
and
σ
Besides the residual error between the model and experimental values,
Table 17.1
also
shows the photoelastic constant
C
5
A
2
B
and the shear stress
τ
in the prism determined as
j
(
σ
1
2
σ
2
)/2
j
. Remarkably, the residual error on emerging wave angles is always smaller
than 0.4%.
The above developments provide a proof of the consistency of the proposed model.
A formal derivation of the fringe formation process is omitted for the sake of brevity.
Table 17.1: Results of Optimization Runs Carried Out to Analyze the Effect of Prism's
Artificial Birefringence
Parameters
Run A
Run B
Run C
n
o
1.5716
1.5709
1.5699
n
e
1.5447
1.5447
1.5468
n
g
1.5399
1.5399
1.5405
A (m
2
/N)
10
2
11
10
2
11
10
2
11
0.5167
0.3017
0.6994
2
3
2
3
2
3
B (m
2
/N)
10
2
11
10
2
11
10
2
11
2.8096
2.9645
3.1390
2
3
2
3
2
3
B (m
2
/N)
10
2
11
10
2
11
10
2
11
C
5
A
2
2.2929
3
2.6628
3
2.4396
3
σ
1
(MPa)
2
135.30
2
142.50
2
161.85
σ
2
(MPa)
2
145.01
2
145.83
2
164.90
τ
(MPa)
4.855
1.666
1.528
Residual error
Ψ
(%)
0.387
0.273
0.339
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